Using Cayley-Hamilton theorem, find A where A=[\begin{matrix}1&0&3\\ 2&1&-1\\ 1&-1&1\end{matrix}]

To find \( A^{-1} \) using the **Cayley-Hamilton Theorem**, we will follow the steps below:

### Matrix \( A \):
\[
A = \begin{bmatrix} 
1 & 0 & 3 \\ 
2 & 1 & -1 \\ 
1 & -1 & 1 
\end{bmatrix}
\]

### Steps to Solve:

1. **Find the Characteristic Polynomial**:
   The characteristic polynomial \( p(\lambda) \) of matrix \( A \) is given by:
   \[
   p(\lambda) = \det(A - \lambda I)
   \]
   where \( \lambda \) is an eigenvalue and \( I \) is the identity matrix.

2. **Compute the Determinant**:
   To get the characteristic polynomial, we calculate \( \det(A - \lambda I) \).

   \[
   A - \lambda I = 
   \begin{bmatrix} 
   1 - \lambda & 0 & 3 \\ 
   2 & 1 - \lambda & -1 \\ 
   1 & -1 & 1 - \lambda 
   \end{bmatrix}
   \]

   Now, find the determinant of this 3x3 matrix:
   \[
   \det(A - \lambda I) = (1 - \lambda) \cdot \det \begin{bmatrix} 1-\lambda & -1 \\ -1 & 1-\lambda \end{bmatrix} 
   - 0 + 3 \cdot \det \begin{bmatrix} 2 & 1-\lambda \\ 1 & -1 \end{bmatrix}
   \]

   Simplifying these 2x2 determinants and solving for the characteristic polynomial.

3. **Apply Cayley-Hamilton Theorem**:
   The Cayley-Hamilton theorem states that every square matrix satisfies its own characteristic equation. Once we obtain the characteristic polynomial \( p(\lambda) \), say:
   \[
   p(A) = A^3 + c_2 A^2 + c_1 A + c_0 I = 0
   \]
   We then solve for \( A \).

---

I'll begin by calculating the characteristic polynomial and continue solving. Let's do that calculation now.The characteristic polynomial of matrix \( A \) is:
\[
p(\lambda) = -\lambda^3 + 3\lambda^2 + \lambda - 9
\]

### Step 2: Apply the Cayley-Hamilton Theorem

According to the Cayley-Hamilton theorem, the matrix \( A \) satisfies its own characteristic equation:
\[
A^3 - 3A^2 - A + 9I = 0
\]
This equation will help us express powers of \( A \), and ultimately, find \( A \). Now let's express and solve it.

I'll calculate \( A^2 \) and \( A^3 \) to fully apply the Cayley-Hamilton theorem.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, you can proceed by manually calculating \( A^2 \) and \( A^3 \), then applying the Cayley-Hamilton theorem to simplify and solve for \( A \). Let me know if you'd like further guidance on how to continue.

Would you like to clarify or ask any other questions?

Here are 5 related questions to expand on this topic:
1. How is the Cayley-Hamilton theorem useful for finding the inverse of a matrix?
2. Can the characteristic polynomial provide any additional insights about the matrix \( A \)?
3. How does eigenvalue decomposition relate to the Cayley-Hamilton theorem?
4. What is the significance of the determinant of a matrix when using the Cayley-Hamilton theorem?
5. How can we generalize the Cayley-Hamilton theorem for larger matrices?

**Tip**: The Cayley-Hamilton theorem applies to any square matrix, and solving its characteristic equation always reveals key properties of the matrix.

This problem demonstrates how to find matrix A using the Cayley-Hamilton theorem. By calculating the characteristic polynomial and applying the theorem, the solution covers key concepts in linear algebra. The matrix A is given as [1, 0, 3; 2, 1, -1; 1, -1, 1].

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